There are three groups of factors that we are interested in; volatility based factors, trading strategy based factors, and macroeconomic factors. For traded factors, we consider SMB, HML, WML, and LIQ. The volatility factors are motivated by Chen (2003). Ang, Hodrick, Xing, and Zhang (2006) and Adrian and Rosenberg (2008) empirically show that the aggregate volatility risk is strongly priced across assets. We use two component volatility model, the GARCH-MIDAS model with rolling window RV, to estimate short- and long-run volatility components. From the innovations to these series, we construct short- and long-run volatility factors.
Chen, Roll, and Ross (1986) proposed a linear factor model of macroeconomic variables. The macroeconomic variables investigated as factors are of MP, UI, DEI, UTS, and UPR.33 They argue that these macroeconomic variables as they are defined
in the paper are noisy enough to be treated as innovations. However, our VAR analysis show that these variables are quite predictable, and hence we take innovations to these variables from the VAR model as our base factors rather than the variables themselves. Shanken and Weinstein (2006) revisit the model and raise concerns about the lack of
32See Cochrane (2001) for more detail arguments. 33Chen, Roll, and Ross (1986) argue that, since IP
tis the flow of industrial production during month t, MPtmeasures the change in industrial production lagged by at least a partial month. To make this
variable contemporaneous with other series, their statistical work lead it by 1 month and we folllow this convention.
robustness of the results in Chen, Roll, and Ross (1986). Chen, Roll, and Ross (1986) use 20 size-sorted portfolios as their test assets, but Shanken and Weinstein (2006) find that their results are surprisingly sensitive to the specific way in which the portfolio returns are generated andβ’s are estimated. Also, Shanken and Weinstein (2006) obtain strong evidence of pricing only for MP and market factor. Our results confirm their concerns and results.
The predictability analyses in Section 4.3 and 4.4 imply that fτ, fg, fW M L, and
fU T S should be strongly priced among all the factors since the variables that generated
first three are very strong predictors of stock market volatility and UTS predicts future market return very strongly. SMB, LIQ, MP, and PPI seem to contain fair amount of information about future market volatility and SMB future market return. However, HML and UPR seem to have no link to either future market returns or future market volatility. The implications from these predictability analyses will be thoroughly investigated in the following horse race of the factors. However, before we get into the horse race, we would like to check how each of these three factor groups performs on our 40 test assets in the first place.
To test the explanatory power of factors chosen, we use the Fama and MacBeth (1973) cross-sectional regression. Our test assets are four different decile portfolios sorted by size, book-to-market ratio, momentum, and liquidity. The monthly returns on these portfolios from July, 1966 to December 1999 are used for the cross-sectional studies. Section 4.1 discusses these portfolios as test assets in detail.
First, we specify a time-series regression that provides estimates of the assets’ loadings: Ri,t=ai+ S X s=1 βi,sFts+ei,t ∀i∈ {1, . . . , I} (20)
where Ri,t is the simple return of asset i at the end of month t, and {Fs}s=1,...,S is a
group of factors chosen from base factors or orthogonalized factors. We obtain full- 39
sample β’s by running the above time-series regression for the full sample. The second step of the Fama-MacBeth procedure is to run a cross-sectional regression with estimated full-sample β’s: Rei,t= S X s=1 Λsβˆi,s+αi,t ∀t∈ {1, . . . , T} (21) where Re
i,t is the return on asset i in excess of the risk-free rate at the end of month
t and Λs’s are the prices of risk for the factor s. A couple of things should be noted.
One is that we impose the null that average pricing error is zero by not including a constant in equation (21). The other is that we are running time-series regressions with simple returns on assets while cross-sectional regressions with excess returns on assets. Since risk-free rates are predetermined ahead of time (and that’s why they are risk-free rate), β’s of returns on the risk-free asset should be all zero. However, if we run time- series regression with asset returns in exess of risk-free rate, the estimated β’s will be contaminated from the spurious non-zero βi,f’s.34
5.1.1 Volatility Component Factors
We start with a set of market and volatility factors,{Re
m,t, ftτ, f g
t}. Table 6 and Table
7 show these results. The volatility factors can be generated from the VAR models; one with y[1]t and the other y[2]t . The results of the former is presented at Table 7 (i), and those of the latter at Table 7 (ii). For a both variable choices, the estimated prices of risk show huge differences across factors in magnitude.35 This can be understood
by looking at the estimated β’s in Table 6 where βτ’s are big and βg’s are very small;
the volatility risk premiums calculated as β ·Λ will be in comparable magnitude with those from market risk. All the prices of risk estimated are turned out to be significant in terms of t-stats computed with Jagannathan and Wang (1998) corrected standard
34For detailed arguments, see Appendix C.
errors.36 To obtain asymptotic covariance matrix of Jagannathan and Wang (1998), we
also used Newey and West (1987) adjustment for autocorrelation.
For overall performance measures for a given choice of factors, we provide three statistics; cross-sectional R2, root mean squared pricing error (henceforth RMSPE), and
pricing error decomposition. When full-sample β’s are used in the Fama-MacBeth cross- sectional regression, the estimated prices of risk are identical to those obtained from cross-sectional OLS regression where mean asset returns are regressed on β’s. Hence, we report cross-sectional R2 as defined in a cross-sectional OLS regression since it is an
informative summary statistic which reflects how well the model fits the data. Although it is a very intuitive measure, it should be interpreted with caution. Jagannathan and Wang (1996) point out that a low R2 does not necessarily indicate that a particular
specification is bad in any absolute sense. Also, Lewellen, Nagel, and Shanken (2006) warn that the high R2’s reported in the literature aren’t nearly as impressive as they
might appear. With simulated artificial factors, Lewellen, Nagel, and Shanken (2006) show that the power of the test is extremely small for three or five factors; the sampling distribution of the adjusted R2 is almost the same when the true R2 is zero and when it
is as high as 70% or 80%.
RMSPE is defined aspP
i( ˆαi)2/Iwhere ˆαi =
P
tαˆi,tand ˆαi,tis the fitted error from
cross-sectional regression in (21). It measures how big the average pricing errors are. The advantage of using this measure of pricing errors and four different decile portfolios as test assets is that we can compute the pricing error decomposition (henceforth PED) as shown in Table 7. The numbers shown as pricing error decomposition are the ratios of sum of squared pricing errors (henceforth SSPE, P
i( ˆαi)2) that belong to a certain decile
portfolios to the total SSPE. Hence, the numbers in pricing error decomposition always
36Shanken (1992) shows how to take into account the sampling errors in the β’s obtained in the
first stage under the assumption that, given the realization of factors, asset returns show conditional homoskedasticity. Jagannathan and Wang (1998) extend Shanken (1992) and derive asymptotic distribution of the estimators without assuming conditional homoskedasticity.
sum up to one. Hence, we can recover SSPE for any decile portfolios. Say, SSPE(size)= RMSPE2×I× PED(size).
Back to Panel A of Table 7, all the measures for the performance of two volatility factor models are very close showing that ftτ[1] and ftg[1] are essentially the same asftτ[2] and ftg[2]. Note that prices of risk for fτ and fg are both significant and both negative,
supporting the empirical results of Adrian and Rosenberg (2008). The estimated β’s from model (ii) of Table 7 are listed in Table 6 and βτ’s are especially well-aligned
with average asset returns of momentum- and liquidity-sorted decile portfolios. In case of momentum portfolios, βτ’s are monotonically decreasing except for the #9 and #10
portfolios. Except for #5, #7, and #8 portfolios,βτ’s are also well lined up with liquidity
portfolios.
Figure 5 and 6 plot expected returns fitted by volatility factor model as specified in (ii) of Table 7 against their realized average returns. Figure 5 and 6 present the same thing. However, since we use four different one-way sorted decile portfolios, we can plot fitted expected returns against realized average returns on each decile portfolios separately as in Figure 6 and we call it a ‘disaggregative view.’ These figures also show fitted pricing errors, ˆαi, in very intuitive way; for each portfolio plotted,
the vertical distance to the 45 degree line is the average pricing error. Pricing error decomposition in Table 7 and Figure 6 give us very intuitive ideas about how the given factor model performs over a certain decile portfolios sorted by economically interesting characteristics. The pricing error decomposition shows that our volatility factor model performs poor on a decile portfolios sorted by book-to-market ratio because almost the half (46%) of total SSPE comes from these portfolios. Figure 6 confirms this. The pricing error decomposition also suggests that the volatility factor model performs fairly poor on momentum portfolios. However, Figure 6 tells us a different story. It turns out that most of the pricing errors come from extreme portfolios on the right (#10). The squared pricing errors of #10 momentum portfolio (( ˆα10)2) is more than the double the
sum of the squared pricing errors of all the rest (#1-#9). In fact, the volatility model prices momentum portfolios very well except for the portfolio #10.
Various specifications in Panel B of Table 7 are to compare fτ t and f
g
t.37 Although
both factors are significantly priced in Panel A, Panel B shows interesting results. t-stat for ftg in model (iv) of Table 7 is fairly strong, but R2 is near zero and RMSPE soars
up to 0.00164. On the contrary, one factor model of fτ
t in (v) shows very impressive
performance. In every measure, the one factor model performs very close to three factor model in (i) and (ii). Figure 7 presents this result. Figure 7 and Figure 6 are essentially the same showing that even pricing errors are similarly distributed.
The Orthogonalized factor specifications in Panel C in Table 7 show many of observations in Panel A and B more clearly. As was discussed in Section 3.2, the order of variables in the VAR model do matter in the construction of orthogonalized factors, and hence we report the order as well in the table. For model (vi) and (vii), the first three variables in y[2]t are ordered as they are shown in Panel C of Table 7 whereas the
rest are ordered in the same way asy[2]t in Section 3.2. Then, we follow Campbell (1996) and Sims (1980) and triangularize the VAR system so that innovations are orthogonal to one another. In the model (vii) of Table 7, the orthogonalization doesn’t affect the market factor in the sense that um,t is still identical to the first element of[2]t . However,
uτt[2] is the corresponding component of the
[2]
t without the common component with
um,t = e01 [2]
t . Similarly u g[2]
t is the corresponding component of
[2]
t orthogonal to both
the market return and the long-run volatility component, and so on. Since these factors are orthogonalized, they might be quite different from the corresponding base factors. This is especially true for uτ
t and u g t because f τ[2] t and f g[2]
t are faily correlated (0.55 in
Table 5). However, three orthogonalized factors in (vi) and (vii) as a group do span about the same space as in (ii) and hence all the performance measures are similar; this is a common feature in all orthogonalized factor specifications. In addition to the
37fτ andfgin Panel B are estimated from VAR model with y[2]
t .
orthogonalization, we rescaled the factors to have the same variance as the innovations to the market return (um,t) and this resolves the problem of drastically varying magnitudes
in estimated prices of risk in (i) and (ii). The model (vi) and (ii) of Table 7 are from the same VAR specification including the same ordering of variables in y[2]t , but the factors in (vi) are orthogonalized while those in (ii) are not. And, this makes a large difference in t-stats of estimated prices of risk. Although ftg is significantly priced in (ii), it loses its significance when it is orthogonalized to the market return and long-run volatility component factor. On the contrary, when the order of τt and gt is reversed in y[2]t as
in Table 7 (vii), i.e. common shocks to τt and gt have been elminated from ftτ but not
fromftg, the long-run volatility component factor loses some of its explanatory power in
cross-sectional variation of mean asset returns but still fairly priced unlikeftg in Table 7 (vi). This is consistent with our VAR predictability results; althoughp-values for both τt and gt are far less than 0.01, that for τt is smaller than that for gt. Being the better
predictor,τtgenerates a factor that encompasses the pricing information ofgtinnovation
factor.
To further examine the information content in fτ
t, or what pricing information
(among SMB, HML, WML, and LIQ)fτ
t captures, we run the Fama-MacBeth regressions
with one factor model of fτ
t on each of decile portfolios separately. For comparison, we
also run cross-sectional regressions with the market factor and the corresponding traded factors that are directly related to the decile portfolios. Note that these traded factors take a form of a spread between two end portfolios (say, #10 and #1), and they are designed to explain the cross-sectional variation of the corresponding decile portfolios. Table 8 shows these results. In terms ofR2and RMSPE, one factor model offτ
t performs
close to the two factor model with the corresponding traded factors in case of size- and liquidity-sorted decile portfolios.38 For the momentum portfolios, one factor model of
38When we add market factor to fτ
t, they get even closer to the two factor models on the left hand
fτ
t achieves R2 = 0.80, but RMSPE is huge when compared with that achieved by
the corresponding two factor model. As is pointed out previously, this is due to the large mispricing of #10 momentum portfolio; if we eliminate #10 portfolio, and test on remaining 9 portfolios, RMSPE of two traded factor model reduces to 0.0003574 while that of fτ
t model drastically reduces to 0.0004866 from 0.0011503. These results are
also consistent with pricing error decomposition for model (i) and (ii) in Table 7. All these empirical evidences suggest that our fτ
t factor summarizes pricing information in
SMB, WML, and LIQ. In a similar sense, we have to conclude that we cannot link HML to stock market volatility.39 These results are largely consistent with our results in the
predictability tests shown in Table 2 and 3; each of {τt, ftSM B, ftW M L, f LIQ
t } is shown
to predict future stock market volatility. The asset pricing equation (17) implies that these factors should be priced across assets. Moreover, since τt is the strongest predictor
of future stock market volatility (τt+1), we won’t be surprised to find out that ftτ wins
the horse races against these factors. In this context, it is also reasonable to find out fτ t
contains pricing information of these traded factors in the set. What is also interesting in Table 8 is that the prices of risk for single fτ
t factor model estimated over each of
decile portfolios are strikingly similar to one another although they are all estimated separately. This explains why one factor model offτ
t performs well over total test assets
of 40 portfolios.
5.1.2 Trading Strategy Based Factors
Table 9 shows results of cross-sectional regressions with trading strategy based factors. Panel A of Table 9 present the results with base factors. Note that the base factors of traded factors are by no means modified or corrected; they are in their original
39The predictability results show that HML does not contain the information about the future market
volatility, and the our cross-sectional studies show that the volatility factor does not contain pricing information of HML as well. It is possible that the linkage to HML got lost in the simplification ofViη,t
in (4). HML might relate to the covariance of consumption growth and market return although this link is not examined in the paper.
form from the data decribed in Section 4.1. Model (i) in Table 9 follows Fama and French (1993) three factor model specification and miserably fails to explain cross-sectional variation of mean asset returns with majority (93%) of SSPE incurred by momenum and liquidity decile portfolios; note that the momentum and the liquidity factors are missing from the model. Three factor models of the market factor and two of any traded factors (SMB, HML, WML, LIQ) show performance that are not significantly different from (i) in Table 9; {Re
m,t, ftHM L, ftW M L} performs the best among those with
R2 = 0.19 and RMSPE=0.001506. All the configurations of the three factor models
share a common feature that most of SSPE incurred by two decile portfolios which are related to the other two missing traded factors.
Carhart (1997) suggests a four factor model as specified in (ii) in Table 9. Table 9 (iii) adopts ftLIQ instead of fW M L
t . In fact, these two models are the best performers
among other possible configurations of four traded factor models and these models perform roughly similar to the one factor model of fτ